What I have now is: Let any sequence $\{$$u_n$} in the Null Space $N(L)$ approach to $u^* \in H$. So the limit of $||Lu_n-u^*|| \leq$ limit of $||u_n-u^*||=0$ for all $L$ as a bounded linear functional in $H$. Then $u^*$ is in $N(L)$ as well. So $L$ is bounded.
Can anyone help me double check if it is correct? Thank you.
Not sure if you just made some typos, but yes the main step is $$0 \le \|Lu\| = \|Lu_n - Lu\| \le \|L\| \|u_n - u\| \to 0.$$ Note the importance of the assumption that $L$ is bounded.